Modeling and Inverse Problems in the Presence of Uncertainty

Authors/Affiliations

H. T. Banks, North Carolina State University, Raleigh, USA

Shuhua Hu, North Carolina State University, Raleigh, USA

W. Clayton Thompson, Noth Carolina State University, Raleigh, USA

This book collects recent research—including the authors’ own substantial projects—on uncertainty propagation and quantification. It covers two sources of uncertainty: where uncertainty is present primarily due to measurement errors and where uncertainty is present due to the modeling formulation itself. With many examples throughout addressing problems in physics, biology, and other areas, the book is suitable for applied mathematicians as well as scientists in biology, medicine, engineering, and physics.

Key Features Reviews basic probability and statistical concepts, making the book self-

contained Presents many applications and theoretical results from engineering, biology,

and physics Covers the general relationship of differential equations driven by white noise

(stochastic differential equations) and the ones driven by colored noise (random differential equations) in terms of their resulting probability density functions

Describes the Prohorov metric framework for nonparametric estimation of aprobability measure

Contains numerous examples and end-of-chapter references to research results, including the authors’ technical reports that can be downloaded from North Carolina State University’s Center for Research in Scientific Computation

Selected Contents Introduction. Probability and Statistics Overview. Mathematical and Statistical Aspects of Inverse Problems. Model Comparison Criteria. Estimation of Probability Measures Using Aggregate Population Data. Optimal Design. Propagation of Uncertainty in a Continuous Time Dynamical System. A Stochastic System and Its Corresponding Deterministic System. Frequently Used Notations and Abbreviations. Index.

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Catalog no. K21506 April 2014, 405 pp.

ISBN: 978-1-4822-0642-5 $89.95 / £57.99

H. T. Banks1, Shuhua Hu1, and W. Clayton

Thompson2

1 Center for Research in Scientific Computation

Department of Mathematics

North Carolina State University

Raleigh, NC

and2 Systems Biology

Cardiovascular, Metabolic, and Endocrine Diseases Research

Unit

Pfizer, Inc.

Boston, MA

Modeling and Inverse

Problems in

the Presence of Uncertainty

CRC PRESS

Boca Raton London New York Washington, D.C.

Contents

Preface xiii

1 Introduction 1

2 Probability and Statistics Overview 32.1 Probability and Probability Space . . . . . . . . . . . . . . . 3

2.1.1 Joint Probability . . . . . . . . . . . . . . . . . . . . . 62.1.2 Conditional Probability . . . . . . . . . . . . . . . . . 7

2.2 Random Variables and Their Associated Distribution Func-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.1 Cumulative Distribution Function . . . . . . . . . . . 92.2.2 Probability Mass Function . . . . . . . . . . . . . . . . 122.2.3 Probability Density Function . . . . . . . . . . . . . . 132.2.4 Equivalence of Two Random Variables . . . . . . . . . 142.2.5 Joint Distribution Function and Marginal Distribution

Function . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.6 Conditional Distribution Function . . . . . . . . . . . 172.2.7 Function of a Random Variable . . . . . . . . . . . . . 20

2.3 Statistical Averages of Random Variables . . . . . . . . . . . 212.3.1 Joint Moments . . . . . . . . . . . . . . . . . . . . . . 232.3.2 Conditional Moments . . . . . . . . . . . . . . . . . . 252.3.3 Statistical Averages of Random Vectors . . . . . . . . 262.3.4 Important Inequalities . . . . . . . . . . . . . . . . . . 26

2.4 Characteristic Functions of a Random Variable . . . . . . . . 272.5 Special Probability Distributions . . . . . . . . . . . . . . . . 28

2.5.1 Poisson Distribution . . . . . . . . . . . . . . . . . . . 292.5.2 Uniform Distribution . . . . . . . . . . . . . . . . . . . 292.5.3 Normal Distribution . . . . . . . . . . . . . . . . . . . 312.5.4 Log-Normal Distribution . . . . . . . . . . . . . . . . . 332.5.5 Multivariate Normal Distribution . . . . . . . . . . . . 352.5.6 Exponential Distribution . . . . . . . . . . . . . . . . 362.5.7 Gamma Distribution . . . . . . . . . . . . . . . . . . . 392.5.8 Chi-Square Distribution . . . . . . . . . . . . . . . . . 412.5.9 Student’s t Distribution . . . . . . . . . . . . . . . . . 42

2.6 Convergence of a Sequence of Random Variables . . . . . . . 442.6.1 Convergence in Distribution . . . . . . . . . . . . . . . 442.6.2 Convergence in Probability . . . . . . . . . . . . . . . 46

v

vi Contents

2.6.3 Convergence Almost Surely . . . . . . . . . . . . . . . 472.6.4 Mean Square Convergence . . . . . . . . . . . . . . . . 49

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 Mathematical and Statistical Aspects of Inverse Problems 533.1 Least Squares Inverse Problem Formulations . . . . . . . . . 54

3.1.1 The Mathematical Model . . . . . . . . . . . . . . . . 543.1.2 The Statistical Model . . . . . . . . . . . . . . . . . . 54

3.2 Methodology: Ordinary, Weighted and GeneralizedLeast Squares . . . . . . . . . . . . . . . . . . . . . . . . . . 563.2.1 Scalar Ordinary Least Squares . . . . . . . . . . . . . 563.2.2 Vector Ordinary Least Squares . . . . . . . . . . . . . 583.2.3 Numerical Implementation of the Vector OLS Proce-

dure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.2.4 Weighted Least Squares (WLS) . . . . . . . . . . . . . 603.2.5 Generalized Least Squares Definition and Motivation . 623.2.6 Numerical Implementation of the GLS Procedure . . . 64

3.3 Asymptotic Theory: Theoretical Foundations . . . . . . . . . 643.3.1 Extension to Weighted Least Squares . . . . . . . . . 70

3.4 Computation of ΣN , Standard Errors and Confidence Intervals 733.5 Investigation of Statistical Assumptions . . . . . . . . . . . . 80

3.5.1 Residual Plots . . . . . . . . . . . . . . . . . . . . . . 803.5.2 An Example Using Residual Plots: Logistic Growth . 823.5.3 A Second Example Using Residual Plot Analysis: Cell

Proliferation . . . . . . . . . . . . . . . . . . . . . . . 873.6 Bootstrapping vs. Asymptotic Error Analysis . . . . . . . . 93

3.6.1 Bootstrapping Algorithm: Constant Variance Data . . 943.6.2 Bootstrapping Algorithm: Non-Constant Variance Data

963.6.3 Results of Numerical Simulations . . . . . . . . . . . . 97

3.6.3.1 Constant Variance Data with OLS . . . . . . 973.6.3.2 Non-Constant Variance Data with GLS . . . 100

3.6.4 Using Incorrect Assumptions on Errors . . . . . . . . 1013.6.4.1 Constant Variance Data Using GLS . . . . . 1033.6.4.2 Non-Constant Variance Data Using OLS . . 103

3.7 The “Corrective” Nature of Bootstrapping Covariance Esti-mates and Their Effects on Confidence Intervals . . . . . . . 106

3.8 Some Summary Remarks on Asymptotic Theory vs. Bootstrap-ping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4 Model Selection Criteria 1174.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.1.1 Statistical and Probability Distribution Models . . . . 1174.1.2 Risks Involved in the Process of Model Selection . . . 119

Contents vii

4.1.3 Model Selection Principle . . . . . . . . . . . . . . . . 1204.2 Likelihood Based Model Selection Criteria – Akaike Informa-

tion Criterion and Its Variations . . . . . . . . . . . . . . . . 1214.2.1 Kullback–Leibler Information . . . . . . . . . . . . . . 1224.2.2 Maximum Likelihood Estimation . . . . . . . . . . . . 1234.2.3 A Large Sample AIC . . . . . . . . . . . . . . . . . . . 1254.2.4 A Small Sample AIC . . . . . . . . . . . . . . . . . . . 126

4.2.4.1 Univariate Observations . . . . . . . . . . . . 1264.2.4.2 Multivariate Observations . . . . . . . . . . . 127

4.2.5 Takeuchi’s Information Criterion . . . . . . . . . . . . 1284.2.6 Remarks on Akaike Information Criterion and Its Vari-

ations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.2.6.1 Candidate Models . . . . . . . . . . . . . . . 1304.2.6.2 The Selected Best Model . . . . . . . . . . . 1304.2.6.3 Pitfalls When Using the AIC . . . . . . . . . 131

4.3 The AIC under the Framework of Least Squares Estimation 1324.3.1 Independent and Identically Normally Distributed Ob-

servations . . . . . . . . . . . . . . . . . . . . . . . . . 1324.3.2 Independent Multivariate Normally Distributed Obser-

vations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1344.3.2.1 Unequal Number of Observations for Different

Observed Components . . . . . . . . . . . . . 1364.3.3 Independent Gamma Distributed Observations . . . . 1384.3.4 General Remarks . . . . . . . . . . . . . . . . . . . . . 140

4.4 Example: CFSE Label Decay . . . . . . . . . . . . . . . . . . 1424.5 Residual Sum of Squares Based Model Selection Criterion . . 146

4.5.1 Ordinary Least Squares . . . . . . . . . . . . . . . . . 1464.5.2 Application: Cat Brain Diffusion/Convection Problem 1494.5.3 Weighted Least Squares . . . . . . . . . . . . . . . . . 1514.5.4 Summary Remarks . . . . . . . . . . . . . . . . . . . . 152

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

5 Estimation of Probability Measures Using Aggregate Popu-lation Data 1575.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1575.2 Type I: Individual Dynamics/Aggregate Data Inverse Problems 160

5.2.1 Structured Population Models . . . . . . . . . . . . . . 1605.3 Type II: Aggregate Dynamics/Aggregate Data

Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . 1635.3.1 Probability Measure Dependent Systems—

Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . 1635.3.2 Probability Measure Dependent Systems—

Maxwell’s Equations . . . . . . . . . . . . . . . . . . . 1675.4 Aggregate Data and the Prohorov Metric Framework . . . . 1695.5 Consistency of the PMF Estimator . . . . . . . . . . . . . . 178

viii Contents

5.6 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 1815.7 Non-Parametric Maximum Likelihood Estimation . . . . . . 181

5.7.1 Likelihood Formulation . . . . . . . . . . . . . . . . . 1825.7.2 Computational Techniques . . . . . . . . . . . . . . . 184

5.8 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 187References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

6 Optimal Design 1956.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1956.2 Mathematical and Statistical Models . . . . . . . . . . . . . 197

6.2.1 Formulation of the Optimal Design Problem . . . . . . 1986.3 Algorithmic Considerations . . . . . . . . . . . . . . . . . . 2026.4 Example: HIV model . . . . . . . . . . . . . . . . . . . . . . 203References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

7 Propagation of Uncertainty in a Continuous Time DynamicalSystem 2097.1 Introduction to Stochastic Processes . . . . . . . . . . . . . . 210

7.1.1 Distribution Functions of a Stochastic Process . . . . 2117.1.2 Moments, Correlation and Covariance Functions of a

Stochastic Process . . . . . . . . . . . . . . . . . . . . 2137.1.3 Classification of a Stochastic Process . . . . . . . . . . 215

7.1.3.1 Stationary Versus Non-Stationary StochasticProcesses . . . . . . . . . . . . . . . . . . . . 215

7.1.3.2 Gaussian vs. Non-Gaussian Processes . . . . 2167.1.4 Methods of Studying a Stochastic Process . . . . . . . 217

7.1.4.1 Sample Function Approach . . . . . . . . . . 2177.1.4.2 Mean Square Calculus Approach . . . . . . . 218

7.1.5 Markov Processes . . . . . . . . . . . . . . . . . . . . . 2207.1.5.1 Characterization of a Markov Process . . . . 2227.1.5.2 The Chapman–Kolmogorov Equation . . . . 2227.1.5.3 An Example of a Markov Process: Wiener

Process . . . . . . . . . . . . . . . . . . . . . 2237.1.5.4 An Example of a Markov Process: Diffusion

Process . . . . . . . . . . . . . . . . . . . . . 2267.1.5.5 An Example of a Markov Process: Poisson

Process . . . . . . . . . . . . . . . . . . . . . 2267.1.5.6 Classification of a Markov Process . . . . . . 2297.1.5.7 Continuous Time Markov Chain . . . . . . . 230

7.1.6 Martingales . . . . . . . . . . . . . . . . . . . . . . . . 2337.1.6.1 Examples of Sample-Continuous Martingales 2347.1.6.2 The Role of Martingales in the Development

of Stochastic Integration Theory . . . . . . . 2347.1.7 White Noise vs. Colored Noise . . . . . . . . . . . . . 236

7.1.7.1 The Power Spectral Density Function . . . . 236

Contents ix

7.1.7.2 White Noise . . . . . . . . . . . . . . . . . . 2387.1.7.3 Colored Noise . . . . . . . . . . . . . . . . . 240

7.2 Stochastic Differential Equations . . . . . . . . . . . . . . . . 2447.2.1 Ito Stochastic Differential Equations . . . . . . . . . . 245

7.2.1.1 Evolution of the Probability Density Functionof X(t) . . . . . . . . . . . . . . . . . . . . . 249

7.2.1.2 Applications of the Fokker–Plank Equation inPopulation Dynamics . . . . . . . . . . . . . 252

7.2.2 Stratonovich Stochastic Differential Equations . . . . 2557.3 Random Differential Equations . . . . . . . . . . . . . . . . . 257

7.3.1 Differential Equations with Random Initial Conditions 2587.3.1.1 Evolution of the Probability Density Function

of x(t;X0) . . . . . . . . . . . . . . . . . . . 2597.3.1.2 Applications of Liouville’s Equation in Popu-

lation Dynamics . . . . . . . . . . . . . . . . 2617.3.2 Differential Equations with Random Model Parameters

and Random Initial Conditions . . . . . . . . . . . . . 2627.3.2.1 Evolution of the Joint Probability Density Func-

tion for (x(t;X0,Z),Z)T . . . . . . . . . . . 263

7.3.2.2 Evolution of Conditional Probability DensityFunction of x(t;X0,Z) Given the Realizationz of Z . . . . . . . . . . . . . . . . . . . . . . 264

7.3.2.3 Applications in Population Dynamics . . . . 2657.3.3 Differential Equations Driven by Correlated Stochastic

Processes . . . . . . . . . . . . . . . . . . . . . . . . . 2687.3.3.1 Joint Probability Density Function of the Cou-

pled Stochastic Process . . . . . . . . . . . . 2697.3.3.2 The Probability Density Function of X(t) . . 273

7.4 Relationships between Random and Stochastic Differential Equa-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2767.4.1 Markov Operators and Markov Semigroups . . . . . . 277

7.4.1.1 Random Differential Equations . . . . . . . . 2797.4.1.2 Stochastic Differential Equations . . . . . . . 281

7.4.2 Pointwise Equivalence Results between Stochastic Dif-ferential Equations and Random Differential Equations 2827.4.2.1 Scalar Affine Differential Equations (Class 1) 2837.4.2.2 Scalar Affine Differential Equations (Class 2) 2857.4.2.3 Vector Affine Systems . . . . . . . . . . . . . 2867.4.2.4 Non-Linear Differential Equations . . . . . . 2887.4.2.5 Remarks on the Equivalence between the SDE

and the RDE . . . . . . . . . . . . . . . . . . 2937.4.2.6 Relationship between the FPPS and GRDPS

Population Models . . . . . . . . . . . . . . . 295References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

x Contents

8 A Stochastic System and Its Corresponding DeterministicSystem 3098.1 Overview of Multivariate Continuous Time Markov Chains . 310

8.1.1 Exponentially Distributed Holding Times . . . . . . . 3108.1.2 Random Time Change Representation . . . . . . . . . 311

8.1.2.1 Relationship between the Stochastic Equationand the Martingale Problem . . . . . . . . . 312

8.1.2.2 Relationship between the Martingale Problemand Kolmogorov’s Forward Equation . . . . 313

8.2 Simulation Algorithms for Continuous Time Markov Chain Mod-els . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3148.2.1 Stochastic Simulation Algorithm . . . . . . . . . . . . 314

8.2.1.1 The Direct Method . . . . . . . . . . . . . . 3158.2.1.2 The First Reaction Method . . . . . . . . . . 315

8.2.2 The Next Reaction Method . . . . . . . . . . . . . . . 3168.2.2.1 The Original Next Reaction Method . . . . . 3178.2.2.2 The Modified Next Reaction Method . . . . 319

8.2.3 Tau-Leaping Methods . . . . . . . . . . . . . . . . . . 3218.2.3.1 An Explicit Tau-Leaping Method . . . . . . 3218.2.3.2 An Implicit Tau-Leaping Method . . . . . . 325

8.3 Density Dependent Continuous Time Markov Chains and Kurtz’sLimit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 3278.3.1 Kurtz’s Limit Theorem . . . . . . . . . . . . . . . . . 3288.3.2 Implications of Kurtz’s Limit Theorem . . . . . . . . . 329

8.4 Biological Application: Vancomycin-Resistant Enterococcus In-fection in a Hospital Unit . . . . . . . . . . . . . . . . . . . . 3318.4.1 The Stochastic VRE Model . . . . . . . . . . . . . . . 3318.4.2 The Deterministic VRE Model . . . . . . . . . . . . . 3338.4.3 Numerical Results . . . . . . . . . . . . . . . . . . . . 334

8.5 Biological Application: HIV Infection within a Host . . . . . 3368.5.1 Deterministic HIV Model . . . . . . . . . . . . . . . . 3368.5.2 Stochastic HIV Models . . . . . . . . . . . . . . . . . . 341

8.5.2.1 The Stochastic HIV Model Based on the BurstProduction Mode . . . . . . . . . . . . . . . 341

8.5.2.2 The Stochastic HIV Model Based on the Con-tinuous Production Mode . . . . . . . . . . . 343

8.5.3 Numerical Results for the Stochastic HIV Model Basedon the Burst Production Mode . . . . . . . . . . . . . 3438.5.3.1 Implementation of the Tau-Leaping

Algorithms . . . . . . . . . . . . . . . . . . . 3438.5.3.2 Comparison of Computational Efficiency of the

SSA and the Tau-Leaping Algorithms . . . . 3468.5.3.3 Accuracy of the Results Obtained by Tau-Leaping

Algorithms . . . . . . . . . . . . . . . . . . . 3488.5.3.4 Stochastic Solution vs. Deterministic Solution 350

Contents xi

8.5.3.5 Final Remark . . . . . . . . . . . . . . . . . 3518.6 Application in Agricultural Production Networks . . . . . . . 352

8.6.1 The Stochastic Pork Production Network Model . . . 3528.6.2 The Deterministic Pork Production Network Model . 3538.6.3 Numerical Results . . . . . . . . . . . . . . . . . . . . 354

8.7 Overview of Stochastic Systems with Delays . . . . . . . . . 3568.8 Simulation Algorithms for Stochastic Systems with Fixed De-

lays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3588.9 Application in the Pork Production Network with a Fixed De-

lay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3598.9.1 The Stochastic Pork Production Network Model with a

Fixed Delay . . . . . . . . . . . . . . . . . . . . . . . . 3608.9.2 The Deterministic Pork Production NetworkModel with

a Fixed Delay . . . . . . . . . . . . . . . . . . . . . . . 3608.9.3 Comparison of the Stochastic Model with a Fixed Delay

and Its Corresponding Deterministic System . . . . . 3628.10 Simulation Algorithms for Stochastic Systems with Random

Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3628.11 Application in the Pork Production Network with a Random

Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3658.11.1 The Corresponding Deterministic System . . . . . . . 3658.11.2 Comparison of the Stochastic Model with a Random

Delay and Its Corresponding Deterministic System . . 3678.11.3 The Corresponding Constructed Stochastic System . . 3688.11.4 Comparison of the Constructed Stochastic System and

Its Corresponding Deterministic System . . . . . . . . 3708.11.5 Comparison of the Stochastic System with a Random

Delay and the Constructed Stochastic System . . . . . 3708.11.5.1 The Effect of Sample Size on the Comparison

of These Two Stochastic Systems . . . . . . 3718.11.5.2 The Effect of the Variance of a Random Delay

on the Comparison of These Two StochasticSystems . . . . . . . . . . . . . . . . . . . . . 375

8.11.5.3 Summary Remarks . . . . . . . . . . . . . . 376References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378

Frequently Used Notations and Abbreviations 383

Index 387

Preface

Writing a research monograph on a “hot topic” such as “uncertainty prop-agation” is a somewhat daunting undertaking. Nonetheless, we decided tocollect our own views, supported by our own research efforts over the past12–15 years on a number of aspects of this topic, and summarize these forthe possible enlightenment they might provide (for us, our students and oth-ers). The research results discussed below are thus necessarily filled witha preponderance of references to our own research reports and papers. Innumerous references below (given at the conclusion of each chapter), we re-fer to CRSC-TRXX-YY. This refers to early Technical Report versions ofmanuscripts which can be found on the Center for Research in Scientific Com-putation website at North Carolina State University where XX refers to theyear, e.g., XX = 03 is 2003, XX = 99 is 1999, while the YY refers to thenumber of the report in that year. These can be found at and downloadedfrom http://www.ncsu.edu/crsc/reports.html where they are listed by year.

Our presentation here has an intended audience from the community of in-vestigators in applied mathematics interested in deterministic and/or stochas-tic models and their interactions as well as scientists in biology, medicine,engineering and physics interested in basic modeling and inverse problems,uncertainty in modeling, propagation of uncertainty and statistical modeling.

We owe great thanks to our former and current students, postdocs and col-leagues for their patience in enduring lectures, questions, feedback and someproofreading. Special thanks are due (in no particular order) to Zack Kenz,Keri Rehm, Dustin Kapraun, Jared Catenacci, Katie Link, Kris Rinnovatore,Kevin Flores, John Nardini, Karissa Cross and Laura Poag for careful read-ing of notes and suggested corrections/revisions on subsets of the material forthis monograph. However, in a sincere attempt to give credit where it is due,each of the authors firmly insists that any errors in judgment, mathematicalcontent, grammar or typos in the material presented in this monograph areentirely the responsibility of his/her two co-authors!!

We (especially young members of our research group) have been Generouslysupported by research grants and fellowships from US federal funding agenciesincluding AFSOR, DARPA, NIH, NSF, DED, and DOE. For this support andencouragement we are all most grateful.

H.T. BanksShuhua Hu

W. Clayton Thompson

xiii

Chapter 1

Introduction

The terms uncertainty quantification and uncertainty propagation havebecome so widely used as to almost have little meaning unless they are furtherexplained. Here we focus primarily on two basic types of problems:

1. Modeling and inverse problems where one assumes that a precise math-ematical model without modeling error is available. This is a standardassumption underlying a large segment of what is taught in many mod-ern statistics courses with a frequentist philosophy. More precisely, amathematical model is given by a dynamical system

dx

dt(t) = g(t,x(t), q) (1.1)

x(t0) = x0 (1.2)

with observation process

f (t; θ) = Cx(t; θ), (1.3)

where θ = (q,x0). The mathematical model is an n-dimensional deter-ministic system and there is a corresponding “truth” parameter θ0 =(q0,x00) so that in the presence of no measurement error the data can bedescribed exactly by the deterministic system at θ0. Thus, uncertaintyis present entirely due to some statistical model of the form

Y j = f(tj ; θ0) + Ej , j = 1, . . . , N, (1.4)

where f (tj ; θ) = Cx(tj ; θ), j = 1, . . . , N , corresponds to the observedpart of the solution of the mathematical model (1.1)–(1.2) at the jthcovariate or observation time and Ej is some type of (possibly statedependent) measurement error. For example, we consider errors that

include those of the form Ej = f(tj ; θ0)γ ◦ Ej where the operation γ◦

denotes component-wise exponentiation by γ followed by component-wise multiplication and γ ≥ 0.

2. An alternate problem wherein the mathematical modeling itself is amajor source of uncertainty and this uncertainty usually propagates intime. That is, the mathematical model has major uncertainties in itsform and/or its parametrization and/or its initial/boundary data, andthis uncertainty is propagated dynamically via some framework as yetto be determined.

1

2 Modeling and Inverse Problems in the Presence of Uncertainty

Before we begin the inverse problem discussions, we give a brief but usefulreview of certain basic probability and statistical concepts. After the probabil-ity and statistics review we present a chapter summarizing both mathematicaland statistical aspects of inverse problem methodology which includes ordi-nary, weighted and generalized least-squares formulations. We discuss asymp-totic theories, bootstrapping and issues related to evaluation of the correctnessof the assumed form of statistical models. We follow this with a discussionof methods for evaluating and comparing the validity of appropriateness ofa collection of models for describing a given data set, including statisticallybased model selection and model comparison techniques.

In Chapter 5 we present a summary of recent results on the estimation ofprobability distributions when they are embedded in complex mathematicalmodels and only aggregate (not individual) data are available. This is followedby a brief chapter on optimal design (what to measure? when and where tomeasure?) of experiments to be carried out in support of inverse problems forgiven models.

The last two chapters focus on the uncertainty in model formulation itself(the second item listed above as the focus of this monograph). In Chapter 7we consider the general problem of evolution of probability density functionsin time. This is done in the context of associated processes resulting fromstochastic differential equations (SDE), which are driven by white noise, andthose resulting from random differential equations (RDE), which are driven bycolored noise. We also discuss their respective wide applications in a numberof different fields including physics and biology. We also consider the generalrelationship between SDE and RDE and establish that there are classes ofproblems for which there is an equivalence between the solutions of the twoformulations. This equivalence, which we term pointwise equivalence, is inthe sense that the respective probability density functions are the same ateach time t. We show, however, that the stochastic processes resulting fromthe SDE and its corresponding pointwise equivalent RDE are generally notthe same in that they may have different covariance functions.

In a final chapter we consider questions related to the appropriateness ofdiscrete versus continuum models in transitions from small numbers of in-dividuals (particles, populations, molecules, etc.) to large numbers. Theseinvestigations are carried out in the context of continuous time Markov chain(CTMC) models and the Kurtz limit theorems for approximations for largenumber stochastic populations by ordinary differential equations for corre-sponding mean populations. Algorithms for simulating CTMC models andCTMC models with delays (discrete and random) are explained and simula-tions are presented for problems arising in specific applications.

The monograph contains illustrative examples throughout, many of themdirectly related to research projects carried out by our group at North CarolinaState University over the past decade.